Theorem eqsb3 2916

Description: Substitution applied to an atomic wff (class version of equsb3 2106). (Contributed by Rodolfo Medina, 28-Apr-2010.)

Assertion

Ref	Expression
eqsb3	⊢ ([𝑦 / 𝑥]𝑥 = 𝐴 ↔ 𝑦 = 𝐴)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem eqsb3

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2802	. 2 ⊢ (𝑥 = 𝑤 → (𝑥 = 𝐴 ↔ 𝑤 = 𝐴))
2		eqeq1 2802	. 2 ⊢ (𝑤 = 𝑦 → (𝑤 = 𝐴 ↔ 𝑦 = 𝐴))
3	1, 2	sbievw2 2104	1 ⊢ ([𝑦 / 𝑥]𝑥 = 𝐴 ↔ 𝑦 = 𝐴)

Syntax hints:   ↔ wb 209   = wceq 1538  [wsb 2069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-cleq 2791

This theorem is referenced by:  pm13.183  3606  eqsbc3  3765